FS0001

signed_square_magnitude_expands

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

The natural magnitude of a normalized signed coordinate squares to the sum of its positive and negative component squares.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall code pos neg. ((code = 2 * pos /\ neg = 0) \/ exists sd_half_fs_square. ((code = 2 * sd_half_fs_square + 1 /\ pos = 0) /\ neg = S sd_half_fs_square)) -> (pos + neg) * (pos + neg) = pos * pos + neg * neg

Constructive proof overview

Generated structural guide

The natural magnitude of a normalized signed coordinate squares to the sum of its positive and negative component squares.

The unchanged tactic script uses 3 declared prerequisites and contains 13 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

signed_decode_normal Alpha theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

13 script commands · 5 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro code
  2. L2
    intro pos
  3. L3
    intro neg
  4. L4
    intro hdecode
02Use earlier factsL5–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L5
    specialize signed_decode_normal code
  2. L6
    specialize signed_decode_normal pos
  3. L7
    specialize signed_decode_normal neg
03Establish hnormalL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed decode normal.

  1. L8
    have hnormal : pos = 0 \/ neg = 0
  2. L9
    apply signed_decode_normal
  3. L10
    exact hdecode
04Separate the logical casesL11–11

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases hnormal
05Calculate and transport equalitiesL12–13

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L12
    simp [hnormal_left, mul_zero_left, zero_add]
  2. L13
    simp [hnormal_right, mul_zero_left, zero_add]

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro code
  2. 0002intro pos
  3. 0003intro neg
  4. 0004intro hdecode
  5. 0005specialize signed_decode_normal code
  6. 0006specialize signed_decode_normal pos
  7. 0007specialize signed_decode_normal neg
  8. 0008have hnormal : pos = 0 \/ neg = 0
  9. 0009apply signed_decode_normal
  10. 0010exact hdecode
  11. 0011cases hnormal
  12. 0012simp [hnormal_left, mul_zero_left, zero_add]
  13. 0013simp [hnormal_right, mul_zero_left, zero_add]